[This week we began work on aggregate demand and the multiplier model.
The lectures go with Case & Fair's Chapter 9.]

LECTURE 28Mon., April 10, 2000

* Today:I. Aggregate demandII. The multiplier model (begin)

I. AGGREGATE DEMAND

New concept:
PLANNED INVESTMENT (Ip): total business expenditures
on plant and equipment (i.e., on capital goods), plus planned production
of inventories (inventory investment)

Recall:
INVESTMENT (I): total business expenditures on plant and equipment,
plus (total) production of inventories.

Ip is not always equal to I

I - Ip = UNPLANNED INVENTORY INVESTMENT, which can be
either positive (unintended inventory accumulation; you produced
too much) or negative (unintended inventory decumulation; you
produced too little and had to run down your inventories)

Recall: GDP (Y), by the product/expenditure approach is:

Y = C + I + G + EX - IM

New concept:
AGGREGATE DEMAND (AD): The total quantity of goods and services
demanded (i.e., purchased).

AD = C + Iplanned + G + EX - IM

-- To repeat: Planned investment does not include "unintended inventory
accumulation". If firms produce goods (say, Edsels) but can't sell them,
those goods are counted in GDP, as (unintended) inventory investment (I),
but they are not part of aggregate demand, because, plainly, nobody was
demanding those goods.
-- The difference between AD and GDP is the difference between planned
investment and total investment. In a word, that difference is inventories.
Unintended inventory accumulation counts total gross investment (I) but
not toward planned investment. Unintended inventory investment can also
be negative (we would call it unintended inventory decumulation),
if AD exceeds output and firms meet the excess demand by selling off goods
that they'd been planning to keep as inventories for the future.
-- AD is often referred to as effective demand, notably
by Keynes.

AGGREGATE EXPENDITURES MODEL: a model in which GDP is ultimately
determined by aggregate demand, and equilibrium GDP is the level of GDP
where aggregate expenditures (AD) equal aggregate production (output).-- also known as the MULTIPLIER MODEL

II. THE MULTIPLIER MODEL

The term multiplier refers to the way that an initial increase
in aggregate expenditures (C, I, G, net EX) causes a ripple effect that
leads to more and more spending and raises GDP by a multiple of
that initial increase in spending. The main reason why this happens is
because when you spend money, the person who receives that money from you
as payment will turn around and spend some of it. And the same thing will
happen when that person spends his money -- the person he paid the
money to will turn around and spend some it, too. The chain of spending
continues until there's nothing left to spend.

Key concept: the marginal propensity to consume (MPC) -- the
fraction of an extra dollar of a person's disposable income that the person
will spend on consumer goods.

How does this multiplier work? A hypothetical example:
-- First off, suppose everyone has the same MPC, 0.75
-- I withdraw $100 from my savings acct and spend it all on a leather
jacket
-- Biff, the leather jacket salesman, since he has MPC = 0.75, spends
$75 (on a hat)
-- Cheryl, the hat salesperson, spends 0.75*$75 = $56 (on a puppy)
-- Ralph, the dog breeder, spends (0.75)2*$75 = $42 (on
a haircut)
-- Olga, the hairstylist, spends (0.75)3*$75 = $32 ...
-- and so on. Note that each subsequent amount spent is 75% of the
previous amount. After many more iterations the amount spent will be so
tiny (75% of a fractional cent) that we can forget about it. But by then
the total increase in spending will have been quite large.

Numerically,let's keep track of the total, cumulative increase
in spending that results from an injection of $100 into the spending stream.
We have assumed MPC = 0.75 and that it's the same for everyone.
-- I spend $100 on a leather jacket. The leather jacket vendor spends
$75 (.75*$100) on a hat, and so on...
-> Increase in equilibrium GDP
= Increase in total spending
=
$100
+ (.75)($100)
+ (.75)(.75)($100)
+ (.75)(.75)($100)
+ ...
= $100 * (1 + .75 + .752
+ .753 + ...)
|
(GEOMETRIC SERIES -- converges to a finite number, according to a simple
formula)

= $100 * [1/(1-.75)]
|
4
= $400

Note that in this example .75 is the MPC.

multiplier = 1/(1-MPC)

Note: the multiplier model is a Keynesian economic model -- that
is, it was first proposed by John Maynard Keynes, in The General Theory.
(The book devotes three whole chapters to the marginal propensity to consume
and the multiplier.) The multiplier model is a model of output determination
-- it tells you what the level of output (GDP) will be, based on the behavior
of consumption, planned investment, and the other components of aggregate
demand.

Although in the real world there are several factors that determine
a household's or a society's consumption, this model focuses on just two
-- (1) everyone's basic subsistence needs (food, clothing, shelter, etc.),
which each of us would somehow provide for even if we had no income, by
borrowing or living off our past savings; and (2) income (people consume
more when they have more income to spend). We break those two types of
consumption down into (1)
autonomous consumption and (2) induced
consumption.

C = a + bY
|
b = marginal propensity to consume (MPC)
b = slope of the consumption function when we graph it

(a is autonomous consumption; bY is induced consumption.)

We assume that:
a > 0 (people's
autonomous consumption is some positive number)
0 < b < 1 (the MPC is positive but less than
100% of people's income)

The above equation is a consumption function -- a simple linear
(straight-line) equation that depicts consumption as a function of disposable
income. Since we're assuming no taxes for now, the consumption function
shows consumption as a function of total income, or GDP (Y).
-- [In class I drew a generic consumption function, with a slope
of b and intersecting the vertical axis at a. A nearly
identical picture appears in Case & Fair's Figure 9.4, on page 180.]

Equivalently, we can use the two above equations (Y = C+S and C=a+bY)
to derive a saving function:

A quick review of some high-school algebra: Algebra is just the
use of letters (like x and y) to represent numbers, especially
numbers whose values can vary (we call such numbers variables) or
are unknown. In macroeconomics, the multiplier model is most straightforwardly
an algebraic model, where C represents consumption spending, S represents
savings, Y represents real GDP, etc.

Algebra and geometry naturally go together. In geometry, we often draw
two-dimensional graphs, with a horizontal axis (which we call the x-axis)
and a vertical axis (the y-axis). We would say that such
a graph is in (x, y) space, since any point on the graph
can be written as (x, y), based on the values of x and y
at that point.

Any straight line can be written algebraically as the equation
of a line:

y = b + mx,
where:

x is the independent variable (x does
not depend on y);

y is the dependent variable (y depends on x);

b is the vertical intercept (or "y-intercept" --
it is the value of y at the point where the line crosses the vertical
axis -- when x = 0, y = b);

m is the slope (the change in y that is associated
with a one-unit change in x).

We can then use the value of the slope and the values (or "coordinates")
of x and y at either of those points to find b, by writing out the
slope equation with one of the given points and with the point (0, b),
which corresponds to the vertical intercept (see example below). Then
we have everything we need to write the line in equation form (y = mx
+ b).

-- Ex.: Suppose we have a line that includes the points (2,3) and (5,7).
[I drew the line on the board. Someday I'll include a graph of it in these
notes on the web.] The slope of this line is:

m = (3-7)
/ (2-5)
= (-4) / (-3)
= 4 / 3
= 1.33

Knowing that the slope is 4/3, solve for b by plugging one of
the given points (say, (2,3)) and the point (0, b) into the slope
formula:

So now we have an equation, of the form y = b + mx, for the line
that runs through the two points (2,3) and (5,7). It is:

y = 0.33 + 1.33x

The consumption function, from last time, can be expressed as the equation
of a line:

C = a + bY,

where C is the dependent variable (it goes on the y-axis),
a
is the vertical intercept (and autonomous consumption), b is the
slope (DC/DY, also
the marginal propensity to consume, MPC), and Y is real GDP (equivalently,
output or real income). When the consumption function is given to us in
equation form, it is easy to graph.

The saving function can also be expressed as the equation of a line:

S = -a + (1-b)*Y,

where S is the dependent variable (it goes on the y-axis),
-a
is the vertical intercept (and autonomous saving), 1-b is the slope
(DS/DY, also the
marginal propensity to save, MPS), and Y is real GDP.

-- [We graphed a consumption function and a saving function in class.
See Case & Fair's Figure 9.6 (page 183) for similar examples.]

II. INVESTMENT

Total investment (I) is the sum of planned investment (Ip)
and unplanned investment (Iu).-- Unplanned investment is unintended inventory production.
For example, if a company produces 100 cars, expecting to sell all of them
this year but only sells 50 this year, then the remaining 50 are counted
in GDP as unplanned inventory investment. (The first 50 are counted as
consumption.)

In real life, planned investment is a function of many factors, including
real interest rates, expectations of future profitability (which would
make firms want to expand production), and the current level of production
(the more you produce, the more you wear out the physical capital stock
and need to replace it). At its simplest, however, the multiplier
model assumes planned investment is fixed at some constant level (e.g.,
Ip = 100). Since investment, as a constant, does not depend
on the level of GDP, we say it is a type of autonomous spending,
just like autonomous consumption (and, in later lectures, government spending
and exports and imports).

The level of government purchases of goods and services (G) also
depends on many factors, but for now, to keep things simple, we will assume
it's fixed at zero (anarchy!). We will assume imports and exports,
and hence net exports, are zero (autarky; no foreign trade).

When the economy is in equilibrium, it will also be the case that
saving equals planned investment. This follows from the original equilibrium
condition, AD = Y and the fact that Y = C + S:
-- If AD = Y then, subtracting C from both sides,
AD - C = Y - C

===> C + Ip - C = S (substituting C + Ip for AD
and S for Y-C)

===> Ip = S (or, as it's usually written, S = Ip)

An example of finding the equilibrium Y, with real numbers:

-- Assume no government (G=0) and no foreign sector (EX=IM=0) and that
consumption and investment functions are as follows:
C = 100 + 0.75Y
Ip = 100

Equilibrium condition:
AD = Y (graphs as 45-degree line
from the origin, because y-intercept is 0 & slope is 1. On any
graph where the vertical and horizontal axes are scaled the same, a line
that extends out from the origin and has a slope of 1 will be a 45-degree
line.)

-- [In class I drew the functions AD = C + Ip + G + EX
- IM and AD=Y. The point where they intersect corresponds to equilibrium
GDP, because at that point AD=Y. The lower graph in Case & Fair's
Figure 9.8 (page 187) is very similar.]

The multiplier is probably most easily calculated as 1/(1-MPC).
As long as you recall that in a consumption function, C = a + bY,b
is
the MPC, then the computation is straightforward:
-- Ex.: C = 100 + 0.75Y
|
MPC = 0.75

multiplier
= 1/(1-MPC) = 1/(1-0.75) = 1/0.25 = 4

More generally and more realistically, investment and import spending
would also depend on the level of income, as might the government's spending,
which historically has risen as GDP has risen. In that case we would also
speak of a marginal propensity to invest, a marginal propensity
to import, and the government's marginal propensity to spend.
And the multiplier would be equal to